Mathematical Research Seminar - Archive

We will present solutions to several problems published in recent volume of ILAS publication IMAGE. Problems are intended to popularize the field of Linear algebra.

http://www.ilasic.org/IMAGE/

In the past we have been using flag graphs and symmetry type graphs for studying certain operations on maps and their symmetries, such as truncation, dual, medial, chamfering, etc. Here we study similar operations on oriented maps. Certain new operations that can only be defined for oriented maps, such as snub, are considered. In the oriented case the primary tools, i.e. flag graphs and symmetry type graphs, are substituted by arc graphs and oriented symmetry type graphs. Same ideas apply to operations on maniplexes and oriented maniplexes and henceforth to polytopes and oriented polytopes. In particular, it is shown that the "twist" operation that Steve Wilson introduced in his talk in Veszprem is naturally defined on oriented maniplexes rather than orientable ones. The talk will be presented using Sage. This is work in progress with Leah Berman,  Gordon Williams and Steve Wilson.

The objective of this talk is to provide an introduction to several open problems about the monodromy groups of abstract polytopes and how they relate to the study of coverings of abstract polytopes by regular abstract polytopes.

Abstract polytopes are a generalization of convex polytopes to purely combinatorial objects, as partially ordered sets satisfying some relatively straightforward constraints. The most symmetric type of abstract polytopes, the regular abstract polytopes, have been studied extensively, and their analysis has been greatly facilitated by the study of their automorphism groups. However, non-regular abstract polytopes are poorly understood, in part because their automorphism group doesn't give enough information about their structure. To begin to investigate non-regular polytopes, we have used two main tools: representation of non-regular polytopes as quotients of regular polytopes, and analysis of the "monodromy group" of the polytope, which encodes combinatorially the connective structure of the polytope. There are important relationships between these two representations, and understanding the relationship motivates many interesting problems in the theory of non-regular abstract polytopes.

This talk will assume no background understanding of either abstract polytopes or monodromy groups. I will provide basic definitions and examples to illustrate both important basic facts about what we have learned about quotient representations and monodromy groups, as well as facilitate introducing open problems in the area.